Mapping of the Late Miocene sandstonefacies using indicator krigingK. Novak Zelenika, T. Malviæ and J. Geiger

PRELIMINARY COMMUNICATION

Facies mapping is one of very important tasks in modelling of oil and gas reservoirs. Facies type has directinfluence on porosity and permeability values, which eventually influence both the migration andaccumulation of hydrocarbons. The most numerous reservoirs in the Croatian part of the Pannonian basinmajor are in the Late Pannonian and Early Pontian sandstones. Various types of these sandstones wereformed in turbiditic depositional environment, which had been periodically activated in relatively calm,deeper (mostly up to 200 meters), brackish lake environment with marl sedimentation over the basin plain.Sandstones form sedimentary bodies that are very elongated in approximately NW-SE direction, with sharptransition toward basin marls in bottom and top. On the contrary, lateral transition is gradual, from theclean, medium-grained sandstones, toward fine-grained sandstones or siltites, silty sandstones, marlysandstones, sandy marls and eventually basin marls. Such lateral facies transition in the Kloštar field wasanalysed, focusing at the largest sandstone oil reservoir ’T’ of Early Pontian age. There were available 19wells with the newest e-logs and calculated average porosity in reservoirs. With 6 additionally constructedvirtual wells using Surfer residual calculation, this made a reliable input dataset. The (litho)facies areanalysed through e-logs, porosity map and, eventually, indicator variograms and Indicator Kriging faciesmap. Transition of porosity values and their probabilities are clearly recognized on the Indicator Krigingmaps, and can be correlated with the interpreted depositional environment at the specific well location. Thisis the first time that Indicator Kriging was applied in Croatian sandstone hydrocarbon reservoirs and resultpositively confirmed that this interpolation technique is appropriate and useful tool for facies mappingbased on subsurface data.

Key words: facies, sandstone, marl, indicator variogram, indicator kriging, Late Miocene, Savadepression, Croatia

1. INTRODUCTION

Mapping of the Upper Miocene depositional facies in theCroatian part of the Pannonian basin is still one of themost important tasks in the Croatian petroleum geology.

Mapping of facies is important for getting insight to sed-imentary environments, shapes and boundaries of hy-drocarbon reservoirs. It is impossible to show allvarieties on one map and that is the reason why severalfacies mapping methods were developed. One of the wellknown methods is based on spontaneous potential (SP)and resistivity (R) curve (in Croatia it was represented inworks of e.g. refs.12, 13, 14, 15, 16, 17

However there is another way for lithofacies interpreta-tion based on subsurface data: if quantitative well-log in-terpretations of porosity are available, specific porosityintervals can be related to some lithofacies types. That ishow by mapping of these intervals the lateral distributionof the corresponding lithofacieses can be predicted. Un-fortunately this is burdened with significant uncertain-ties. On one hand, the calculation of effective porosityfrom well-logs has uncertainty of its own. Consequentlysometimes the meaningful question is not ’What is theporosity at a particular location’, but rather ’What is theprobability of porosity lying in a particular interval’ or’What is the probability of porosity being smaller/largerthan a particular cutoff ’. On the other hand, there hasn’tbeen any traditional way to map the lateral distributionof a porosity interval for a long time.

However, by continuous development of geostatisticaltools, during the late 80’s a special method for krigingwas introduced, called Indicator Kriging (IK).5, 7 Thismethod has already been applied by other authors2, 3, 4, 7

to address the problems mentioned above. Later, severalauthors applied this method for a wide variety of prob-lems including mapping of soil types2 and facies map-ping8 as categorical variables, estimating of geological at-tributes with great number of extreme values,9 or map-ping of the production data of a hydrocarbon reservoir.6

By accumulating the practice, several textbooks ap-peared, summarizing these applications and giving theo-retical background of IK.

In this paper certain key questions are summarizedconcerning the meaningful application with aim to dem-onstrate the possible application in approachinglithofacies mapping using IK of porosity from well-logs.An old oil field located close to the margin of the Sava de-pression named Kloštar field is chosen as a suitablestudy area.

2. THE STUDY AREAMost of the oil and gas reservoirs in SW corner of thePannonian basin (i.e. in the Sava, Drava, Slavonia-Srijemand Mura depressions) are in sandstones of LatePannonian and Early Pontian age (Late Miocene).Depositional model of these sandstones is well inter-preted by now.10, 12, 14, 16, 17 They represent a result of peri-odical activity of turbiditic currents in calm, brackish,

NAFTA 61 (5) 225-233 (2010) 225

lake environment, where hemipelagicmarls were deposited most of the time.

Late Miocene sandstone-proneturbidites entered the structural de-pressions, located between regionalstrike-slip faults (mostly like pull-apartbasins). Parts of these depressionswere later subjected to structural up-lifts as a result of the inversion due totranspression resulted from the N-Sorientation of the horizontal compo-nent of regional stress. Hydrocarbonsstarted to migrate in these traps in LatePliocene and Early Quaternary, andprobably even in Late Miocene (e.g. ref.1).

Sedimentation in smaller structuraldepressions had great influence on fa-cies distribution, as well as on bound-aries of the reservoir. Fault planes arealso often boundaries betweenturbiditic sandstone and basin marl. Itmeans that sandstones had been de-posited periodically, with marl in theirtop and base. Lateral facies boundariesare not sharp and there is wide transi-tion zone toward the basin plain. Later-ally there is transition from (in axialparts of submarine channels) me-dium-grained to fine-grained sand-stone, marly sandstone, sandymarlstone and finally marl. Also,granulometric composition of sand-stones changes in the direction ofturbidite current palaeotransport,10

from medium-grained to fine-grainedsandstones.

Kloštar field is located 35 km east ofZagreb on the western slopes ofMoslavaèka Gora Mountain, with alti-tude from 110 to 180 m.

Geologically, Kloštar field belongs tothe Sava depression. Reservoir rocksare of Miocene age and Palaeozoic age.There are 5 oil and gas pronelithostratigraphic units with altogether20 reservoirs. Sandstone reservoirs arelocated in three units. The oldest isnamed ’Pre-Valencianensis beds’ (EarlyPannonian) with sandstone lens. Reser-voirs of the ’2nd sandstone series’ and’1st sandstone series’ (Early Pontian)have greater lateral extension and con-tain major hydrocarbon reserves.

3. DATA SET

The input dataset included porosity data collected in 25points of Kloštar field, as average value of Lower Pontianoil reservoirs of ’1st sandstone series’.

In the analysed reservoir, porosity from wells varies be-tween 13.8 and 23.3%. It is assumed that the lowest

value represents marlitic sand, as the transitional facies

toward basin marl. On contrary, the maximal value had

been measured in the clean sandstone, deposited in the

deepest and central part of the transport channel. Of

course, the average porosity value of wells strongly de-

pends of their location in depositional environment, and

it is represented by geological sections of the two men-

tioned “extreme” wells shown on Figures 1 and 2.

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K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER MAPPING OF THE LATE MIOCENE SANDSTONE...

Fig. 1. Composite geological column (including stratigraphy, e-logs andlithology) of the well with minimum average porosity in the analysedsandstone reservoir 'T' (location belongs to sandy marl lithofacies)

Sl. 1. Slo�eni geološki stup (ukljuèuje stratigrafiju, karota�u i litologiju) kroz bušotinuu kojoj je odreðena najmanja srednja poroznost u analiziranom le�ištu 'T'(lokacija pripada litofacijesu pjeskovitog lapora)

4. INDICATOR KRIGING (IK)The following brief review on indicator formalism and In-dicator Kriging technique are based on the works of refs.3, 4, 5, 6, 7

4.1. Indicator formalism and someconsequences

Within an uncored region, one of the most frequentlyused method for facies identification is the usage of po-

rosity cutoffs. More precisely, if a par-ticular facies can be recognized by aícutoff, and if the presence of this faciesis assigned with 1 and absence with 0respectively, one can introduce aso-called indicator variable (like in Fig-ure 3) with the following form:

� �� �

� �I x

z x v

z x v

cutoff

cutoff

��

�

��

1

0

if

if(4.1)

where:

I(x) is the indicator variable;z(x) is originally measured value;vcutoff is cutoff value.

Let’s assume that instead of a singlethreshold ícutoff, l cutoffs íl are appliedto the observed data, such that

� �� �

� �I x v

z x v

z x vl

l cutoff

l cutoff

, ��

�

��

1

0

if

if(4.2)

where z(x) is the value of aregionalized variable such as porosityat location x.

Thus, raw data are transformed into lnew variables, each taking on the valueof 0 or 1. Note, that by selecting vl cut-offs the main purpose is to obtain a rea-sonable picture of the frequency ofwells or reservoir sections above or be-low each cutoff. It can be shown, thatthe probability of porosity z(x) below acutoff íl within an area A equals

� � � �P A zA

i x v dxl

A

; ,� 1

(4.3)

With knowledge of P(A, v), one can de-rive the probability of wells with poros-ity above cutoff vl:

� �� � � �P z x v P A vl� � �1 , (4.4)

This probability can be estimated di-rectly from n observed values of z(x):

� � � � � �P A v v i x xi

i

n

l k l

* , ,��

� 1

(4.5)

where i (i=1, 2, …, n) are n weightsbeing calculated from a kriging systemthrough calculation of residual indica-tor data [i(xk, v)-F*(z)].7 In this latter ex-

pression F*(z) is an unbiased estimate of frequency, F(z):

� � � �� �F z E P A v� , (4.6)

One of the most impressive consequences of the abovederivations is that the aim of the indicator formalism forcontinuous variables is to directly estimate the distribu-tion of uncertainty at unsampled location. The globalprobability distribution function of the input data setalso then can be estimated at a series of threshold values.

MAPPING OF THE LATE MIOCENE SANDSTONE... K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER

NAFTA 61 (5) 225-233 (2010) 227

Fig. 2. Composite geological column (including stratigraphy, e-logs andlithology) of the well with maximum average porosity in the analysedsandstone reservoir 'T' (location belongs to medium grained sandstoneslithofacies)

Sl. 2. Slo�eni geološki stup (ukljuèuje stratigrafiju, karota�u i litologiju) kroz bušotinuu kojoj je odreðena najveæa srednja poroznost u analiziranom le�ištu 'T'(lokacija pripada litofacijesu srednjozrnatih pješèenjaka)

4.2. Brief summary of Indicator Kriging (IK)

If data are not clustered spatially, the estimate of F*(z)can be done from the histogram of all data available. Inthe case of Simple Kriging estimator, the residual [P*(A,vl)-F*(vl)] is used to derive as follows:

� � � � � � � � � �� �P A v F v v i x v F vl l i l

i

n

i l l* , * , *� � ��

� 1

(4.7)

where i(vl) is the k-th weight for cutoff vl.

Note, the simple kriging estimator differs from the Or-dinary Kriging approach by not requiring the sum ofweights to equal one. The Simple Kriging system of equa-tion can be given as:

� � � � � � � �k l

k

n

i i m l k k h lv x x v x x v k K n�

�� � � �1

12; ; , , , , , (4.8)

In the above equation � �� i i m lx x v� ; are the indicator

semivariogram values for the distance xi-xm at cutoff vl,

and the terms � �� x x vk k h l, ;� are average indicator

semivariogram values between location xk and xk+h at cut-off vl.

The IK process is repeated for all l cutoff (threshold)values, which discretize the interval of variability of thecontinuous attribute z. The distribution of uncertainty,built from assembling the l indicator kriging estimates,represents a probabilistic model for the uncertaintyabout the unsampled value z(u). Obviously this IndicatorKriging procedure requires a variogram measure of cor-relation corresponding to each threshold.

It is important to note, that correct selection of the l cut-offs is essential for Indicator Kriging, In case of too many

cutoffs the computation time increase drastically, butwith too few cutoffs one can lose some important detailsof the distribution.3 In general the number of cutoffsshould be between 5 and 11.

Kriging of an indicator variable does not result only invalues 0 and 1, but rather estimates along a continuousscale in the [0,1] interval. Thus Indicator Kriging yieldsprobabilities (or relative frequencies) of the {z(x) < vl}events. Assuming that rank-ordered cutoffs are (v1 < v2

< …< vn), it is obvious that the estimated probabilitiesmust obey the relations:

� �� � � �� �P z x v P z x vl l* ; * ;� � 1 for all l (4.9)

Linear estimation of probabilities belonging to l cutoffspermits one to draw several types of maps. For each cut-off, e.g. a map of the probabilities of not exceeding cutoff(i.e. P*(z(x), vl)), or the probabilities of exceeding vl: (1-P*(z(x), vl) can be evidently drawn. It is also evident thatthe estimated mean can be calculated through a discretesum:

� � � � � �� �q A v P A v P A vl

i

L

l l* ; ; * ;*01

1� ��

�� (4.10)

where vl

* is the central value of the interval � �v vl l�1; .

Isoquantile maps can be drawn from the conditionaldistribution function fitted at each grid node. For in-stance, the median can be found by interpolation be-tween vmax and vmax-1, where P*(A;vmax) is the highest

value of P*(A;v)�0.5. The same procedure yieldsquantiles for any value of p, allowing one to map confi-dence intervals around a mean or median. The estimatedconfidence intervals can be calculated directly from theconditional distribution and do not need any assumptionof the type of distribution of estimation variance.

4.3. Advantages and side effects

Using Indicator Kriging does not require stationarity as-sumption, or multivariate normality. One of the most im-portant advantages is its robustness in respect to theextreme (outlier) values. Another important fact arisesfrom the indicator formalism. By indicator cutoffs theoriginally continuous distribution is discretized. Fromthis point the analysis runs on interval data rather thancrisp data. That is, the preciseness of the input data isnot necessarily required. It is enough to know that at aparticular location the porosity lies in a particular inter-val. If this meets with one of the initially defined cutoffpairs, the input data set can be extended by adding thisinformation. This is the property which we used in thiswork to make the input data set denser.

5. ADDING OF THE NEW “HARD”DATA AND VARIOGRAM ANALYSISOF INDICATOR SET

The input dataset included porosity data collected in 19wells from the reservoir ’T’. The very densely verticallyspaced porosity values were averaged well by well for thewhole reservoir interval. The minimum of 5.448% hasbeen excluded from further analysis because it was re-cognised as outlier.

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K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER MAPPING OF THE LATE MIOCENE SANDSTONE...

Fig. 3. Expression of 35 imaginary values where '1' used forsandstone locations and '0' for non-sandstonelocations

Sl. 3. Skupina od 35 zamišljenih podataka gdje'1' predstavljalokacije pješèenjaka, a '0' lokacije ostalih litofacijesa

The wells are very irregularly spaced, with large areas

of missing information (Fig. 4). To avoid this problem, six

additional points (A, B, C, D, E and F) were set to the lo-

cations with lack of data. Their porosity values were esti-

mated from the neighbouring wells using a linearvariogram model. Additional hard data, their estimatedporosity and X, Y locations are visible also in Figure 4and in Table 1.

The presumption was that porosity values indicate dif-ferent sandstone facies (i.e. marly sand, fine-grained andmedium-grained sandstones). It is why porosity datasethad been transformed in the 6 indicator dataset, basedon following (Table 1) cutoffs: 14, 18, 19, 20, 22 and24%.

For each of the cutoffs, the corresponding indicatorvariogram was calculated using Variowin 2.21.11

The variogram models to be used in indicator krigingalgorithm must obey the following criteria:3

• The theoretical (model) function must be the same(spherical is used here)

• The sill must be identical (standardized variogram)

• The nugget effect must be the same (it is zero here)

• Only the range can change for different indicator vari-ables (all ranges were relatively low).

For keeping the simplicity all the experimental indica-tor variograms are considered as omnidirectional (Fig.5). For each of the six variables, the starting lag spacingwas set to 250 m.

6. INDICATOR KRIGING MAPPINGUSING PROGRAM ’WINGSLIB’

Indicator Kriging (abbr. ’IK’) in WinGslibTM package isprimarily used to generate conditional probabilities

MAPPING OF THE LATE MIOCENE SANDSTONE... K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER

NAFTA 61 (5) 225-233 (2010) 229

Fig. 4. Porosity distribution based on hard data derivedfrom wells (+) and additional artificial hard data (?)with porosity estimated using SurferTM

Sl. 4. Distribucija poroziteta na temelju èvrstih bušotinskihpodataka (+) i dodatno konstruiranih toèaka (?) ukojima je poroznost procijenjena SurferomTM

X Y Por (%) 14% 18% 19% 20% 22% 24%

6376161.26 5067837.14 20.045 0 0 0 0 1 1

6376598.12 5067814.56 20.525 0 0 0 0 1 1

6376734.51 5067596.74 21.163 0 0 0 0 1 1

6376888.03 5068296.06 21.093 0 0 0 0 1 1

6377036.31 5068042.44 23.282 0 0 0 0 0 1

6376967.36 5067600.97 22.036 0 0 0 0 0 1

6377085.97 5068506.80 19.666 0 0 0 1 1 1

6377275.39 5068080.14 19.164 0 0 0 1 1 1

6377192.49 5067820.25 19.499 0 0 0 1 1 1

6377340.05 5067550.55 19.863 0 0 0 1 1 1

6377490.35 5066730.24 18.061 0 0 1 1 1 1

6377589.91 5067642.00 19.617 0 0 0 1 1 1

6377672.02 5066901.68 18.504 0 0 1 1 1 1

6377888.33 5066692.16 18.166 0 0 1 1 1 1

6377821.07 5067850.14 17.939 0 1 1 1 1 1

6377977.74 5066964.14 19.628 0 0 0 1 1 1

6378168.97 5066731.69 21.808 0 0 0 0 1 1

6378263.31 5068258.95 18.363 0 0 1 1 1 1

6378478.38 5067245.16 13.798 1 1 1 1 1 1

6376413.48 5068275.11 20.584 0 0 0 0 1 1

6376576.83 5067225.75 20.379 0 0 0 0 1 1

6377165.85 5067027.76 19.593 0 0 0 1 1 1

6377660.83 5067374.25 19.280 0 0 0 1 1 1

6378101.37 5067715.78 17.430 0 1 1 1 1 1

6377715.28 5068240.46 18.513 0 0 1 1 1 1

Table 1. Indicator transformations of porosity based on different cutoffs. Coordinates are in Gauss-Krueger system in zone 5(E13°30' - E16°30) with latitude of origin 0° and longitude of origin 15°.

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Fig. 5 Experimental variograms for different cut-offs (left) and their approximations with theoretical curves (right)Sl. 5 Eksperimentalni variogrami za razlièite graniène vrijednosti (lijevo) i njihova teorijska aproksimacija (desno)

within the ’sisim’ stochastic simulation program. More-over ’ik3d’ is program for simply and ordinary IndicatorKriging for categorical variables or cumulative indicatorscalculated for continuous variable. In the case of continu-ous variable, the Indicator Kriging algorithm provides in-dependent discrete models (discrete probabilities) forthe various cutoffs.

The flexibility of the IK approach comes from modellingthe various probabilities with different variogram dis-tances. Program ’ik3d’ provides the necessary statistics,but with two constraints. The first is related to the esti-

mated value � �� �F u z nk; to the bound 0 or 1, if originally

valued outside the interval [0, 1]. The second constraintinvolves ’k’ separate kriging results and it is harder con-dition than the first one. These constraints can help withnegative indicator kriging weights or lack of data. Figure6 is a map showing well locations and values. All maps inpaper are created with program WinGslibTM.

Cumulative probability distribution curve is obligatoryinput for indicator mapping. On Figure 7 ’X’ axis repre-sents classes and ’Y’ probabilities (in percentages).

Lots of input parameters had to be defined using ’ik3d’.Within the WinGslib system, the executables areparameterized by a special parameter file (abbr. ’par’).The procedure needs to be repetitive and that is why themain parameters of ’ik3d’ program are listed here.

A ’Full IK” approach was used with ’Simple Kriging’ es-timation. This technique was chosen because it is basedon global mean, and input dataset is small to take locallyvarying mean into consideration. In ’Grid definition’ thenumber of cells both in X and Y directions was 251. Thefinal model contains 63 001 cells.

As for cutoffs, six porosity values were selected: 14, 18,19, 20, 22 and 24%. The corresponding cumulativeprobabilities were 0.08, 0.17, 0.36, 0.66 and 0.92respectively.

The so called “E-type estimation” of porosity averagedfor the whole reservoir interval is shown in Figure 8. Thisshows the most probable porosity value in the formationanalysed.

Finally, the interpolated probability maps, for each ofthe selected cutoffs are shown in Figure 9.

Lateral distribution of the all mentioned facies (sand-stone, marly sandstone, sandy marlstone and marl)across the analysed reservoir can be observed. Ofcourse, such distribution depends on the selected cut-offs. In this case, the lowest porosities came from thepart of reservoir with low oil saturation, close to theoil-water contact. The highest porosities correspond tothe best part of the reservoir, i.e. these values are mea-sured in the middle of the sandstone body.

The probability maps (Figure 9) have coloured legend.The blue colour means that there is no probability that

MAPPING OF THE LATE MIOCENE SANDSTONE... K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER

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Fig. 6. Hard-data value map of 19 wells in Kloštar field withporosity value on each well, as well as 6 additionalartificial hard data

Sl. 6. Polo�ajna karta "èvrstih" (mjerenih) vrijednostiporoznosti u 19 bušotina na polju Kloštar, kao i 6dodatno konstruiranih toèaka

Fig. 7. Cumulative probability distribution for 8 porosity classes(Axes: X are porosity classes, Y is cumulative probabilityin %)

Sl. 7. Krivulja kumulativne vjerojatnosti prikazana kroz 8razreda poroznosti (na osi X su vrijednosti razreda, naosi Y kumulativne vjerojatnosti u %)

Fig. 8. E-type estimation of porosity derived from IKestimations of 5 cut-offs (14%, 18%, 19%, 20%, 22%)

Sl. 8. E-tip procjena poroznosti izraèunata iz karata IK za 5graniènih vrijednosti (14%, 18%, 19%, 20%, 22%)

the cell value will be less than selected cutoff (p = 0). Thered colour shows that cell value is for sure lower than se-lected cutoff (p = 1).

For example, for the cutoff = 14% (Figure 9) only thecells being in the SE corner of the study area can have(with some probability) porosities lower than 14%. Theprobability of finding such a low porosity values is 0 forall the other parts of the map.

On contrary, the map for cutoff=22% (Figure 9) is al-most entirely in red colour (except the two central loca-tions). It means that almost all cells will surely (p = 1)have the values lower than 22%. This is the way how toread this set of probability maps (Figure 9).

7. DISCUSSION

The Indicator Kriging is a specific geostatistical tech-nique for spatial phenomena with weak stationarity. In

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Fig. 9. Probability maps for porosity cutoffs. Probability '1' means for sure that value of cell will be lesser than selected cut-offSl. 9. Karte vjerojatnosti za graniène vrijednosti poroznosti. Vjerojatnost '1' znaèi kako je sigurno da æe vrijednost æelije biti manja od

odabrane graniène vrijednosti

fact, this kriging technique is weaker than any otherkriging approximation. However, this technique is de-signed for estimating lateral uncertainty rather than’crisp’ values. It means that this approach estimates thelocal probability distributions on grid cells. IK workswith interval data discretized form of the originally con-tinuous distribution of a regionalized variable. That iswhy this method can be used at any case when data aretoo noisy, or when the ’exact’ value is not known, but onecan reliably estimate its interval (range of values).

Looking again at stationarity, there are three level ofthis condition. The first-order stationarity (which is in-variant for any translation) is very strict requirement thatcannot be satisfied using natural dataset. The weakerform is so called second-order stationarity which re-quires that the expected value must be independent onlocations, and needs that the covariance is dependentonly on the separation vector between any two locations.The third form is the so called intrinsic hypothesis whichexpects the mean to be independent and the existence ofsemivariogram. Just this third-order stationarity is as-sumed, because the decision to use variogram modelmeans only the intrinsic hypothesis is expected as theminimum that the dataset has to satisfy (disregarding thenugget model).

Most of the kriging techniques are linear, but some ofthem are not. In fact, these are linear techniques appliedon some non-linear transformation of the data. Indicatortransformation (like in Equation 4.4) presented in analy-sis is one of such non-linear transformation and Indica-tor Kriging is non-linear technique as well. Suchapplication in this analysis resulted in indicatorvariograms for different porosity cutoffs and, more im-portant, in set of probability maps for such cutoffs. Usingof these maps revealed the areal extension of porosityprobability below defined cutoff.

ACKNOWLEDGMENTS

This paper represents the theoretical analysis ofgeomathematical methods performed in 2009 in the pro-ject “Stratigraphical and geomathematical research ofpetroleum geological systems in Croatia (project no.195-1951293-0237)”, financed by the Ministry of Sci-ence, Education, and Sports of the Republic of Croatia.

Porosity data are taken from the project finished in2008 “Improvements in geological interpretation meth-ods with the purpose of increasing recovery in sandstonereservoirs”, financed and supported by INA-Oil CompanyPlc. This was a cooperative endeavour of the Field Engi-neering and Reservoir Development Dept. (INA Plc.,Croatia) and the Institute of Geology and Geological Engi-neering (Faculty of Mining, Geology and PetroleumEngineering, University of Zagreb, Croatia).

We wish to thank author of VARIOWIN 2.21., Mr. YVANPANNATIER, for using one of the most popular freewaresoftware for variogram analysis. Variowin copyright ©1993, 1994, 1995 belongs to Yvan Pannatier.

Maps are created by licensed version of WinGslibTM

software package.

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Kristina Novak Zelenika, INA-Oil industry, Sector for Geology and Reser-voir Management, Šubiæeva 29, 10000 Zagreb (Reservoir Geologist).e-mail: kristina.novakzelenika@ina.hr

Tomislav Malviæ, INA-Oil industry, Sector for Geology and Reservoir Man-agement, Šubiæeva 29, 10000 Zagreb (Adviser).University of Zagreb, Faculty of Mining, Geology and Petroleum Engineering,Pierottijeva 6, 10000 Zagreb (Assistant Professor)e-mail: (Reservoir Geologist); tomislav.malvic@ina.hr

Janos Geiger, University of Szeged, Department of Geology and Paleontol-ogy, Egyetem street no. 2, Szeged, Hungary (Associate Professor).e-mail: matska@geo.u-szeged.hu

MAPPING OF THE LATE MIOCENE SANDSTONE... K. NOVAK ZELENIKA, T. MALVIÆ AND J. GEIGER

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